某些非齐性流形上极大函数和Riesz变换的研究

Maximal functions and Riesz transforms play an important role in classical harmonic analysis. It is an interesting question to study the mapping properties in other settings. In fact, the mapping properties of maximal functions and Riesz transforms on manifolds can be very different from that on homogeneous spaces when these manifolds do not satisfy the doubling conditions. In this project, we will study the mapping properties of maximal functions and Riesz transforms on some non-homogeneous manifolds. Firstly, we consider the direct product manifolds where at least one of the submanifolds grows exponentially. For instance, we consider the product of exponential growth manifold and a manifold satisfying the doubling condition as well as the product of two weighted hyperbolic spaces. On which we study the relations between the mapping properties of centered maximal function and that on submanifolds. Moreover, we consider the operator norms of the centered maximal operators on weighted hyperbolic spaces and try to give the upper bounds with respect of the dimensions. Finally, we consider the properties of Riesz transforms on rotationally symmetric manifolds. Through proper upper bounds of the heat kernel, we can get the Lp boundedness of the Riesz transform. In particular, we consider a class of rotationally symmetric manifolds with Ricci curvature unbounded from below.

极大函数和Riesz变换是经典调和分析中重要的研究对象，一个有趣的问题是研究它们在其它结构上的映射性质。事实上，如果流形不满足倍测度性质，已经有结果表明极大函数和Riesz变换的映射性质与齐性空间上的映射性质情况不同。本项目主要考虑了某些非齐性流形上的极大函数和Riesz变换的映射性质。首先，我们考虑了直积流形，其中至少有一个子流形的体积是指数增长的。例如一个满足倍测度性质的流形和一个体积指数增长的流形的直积流形，两个加权双曲空间的直积流形。我们研究了这些直积流形上中心极大函数的映射性质和子流形上中心极大函数映射性质的关系。另外，我们还考虑了加权双曲空间上中心极大函数的算子范数与维数的关系。最后，我们考虑了旋转对称流形上的Riesz变换的映射性质。通过研究该流形上热核的性质，我们希望建立Riesz变换的Lp有界性。特别的，我们考虑了一类Ricci曲率无下界的旋转对称流形。

本次研究主要是关于流形上的调和分析。具体而言，我们首先考虑了乘积加权双曲空间上的中心极大函数的映射性质。我们证明了，如果乘积流形是由两个维数相同的加权双曲空间构成时，我们可以用权的性质完全刻画中心极大函数的映射性质。另一方面，为了研究旋转对称流形上的Laplace-Beltrami算子的性质，我们研究了薛定谔算子的性质。事实上，我们研究了薛定谔传播子的积分核的连续性。最后，我们还研究了时空分数阶薛定谔方程的适定性。